Waves for AP Physics Alumni

Vocabulary Term symbol meaning (if any) amplitude A how far the particles in the medium move away from their equilibrium positions antinode in a standing wave, a point in the medium that moves the most compression in a longitudinal wave, a point where the medium is squished together (in a gas, this means high pressure) crest the highest point in a wave

frequency f how many times the particles in the medium oscillate per second. Also, the frequency of the SHO that’s driving the wave. fundamental frequency in a standing wave, the lowest-frequency wave that can resonate. This is the first mode so n = 1 . longitudinal wave a wave in which the particles in the medium oscillate parallel (back and forth) to the direction of the wave’s motion medium the substance that the wave travels through node in a standing wave, a point in the medium that doesn’t move period T how long it takes the particles in the medium to go through a complete cycle of oscillation. Also, the period of the SHO driving the wave. rarefaction in a longitudinal wave, a point where the medium is pulled apart (in a gas, this means low pressure) speed v how fast the wave travels (not how fast the particles oscillate) standing wave a wave that is in a finite amount of medium, bouncing back and forth at each end transverse wave a wave in which the particles in the medium oscillate perpendicular (up and down) to the direction of the wave (this is the most typical type of wave) trough the lowest point in a wave wavelength λ

Waves are Interesting (Hewitt 25) A wave is what happens when you connect an oscillation to a medium whose particles are connected to each other in some way. This could be a flexible solid like a string, a liquid, or a gas. The important thing to note is that the wave moves through the medium while the particles just move back and forth close to where they are.

In the picture, you see that the wave is moving to the right, but the points on the rope are just moving up and down. The energy and momentum move to the right as well. This is a picture of a transverse wave. Most waves that we think of are transverse. Waves in strings, waves in water, the “wave” that people do in sports stadiums – they’re all transverse. In transverse waves, the particles of the medium move back and forth perpendicular to the motion. Another type of wave is longitudinal. In this case, they move parallel (and anti- parallel) to the motion. Sound is a longitudinal wave.

If you have a slinky, you can experiment with transverse and longitudinal waves this way. Sound is a longitudinal wave in air. When a sound wave is travelling through air, it causes areas of higher and lower pressure because the air molecules get pulled apart and pushed together.

What We Measure Some of the calculations for waves look a lot like the calculations for the simple harmonic oscillator and circular motion because we use the same quantities: frequency and period. The main important equation is this one: v = λ f v = wave velocity (m/s) λ = wavelength (m) f = frequency (Hz) which makes total sense if you just look at the units. This is probably the most important thing to memorize in this unit.

It is important to notice that the speed of the wave doesn’t depend on the wavelength, the frequency, or the amplitude. The speed is determined only by the properties of the medium. When the medium is stiff (for example a spring with a high spring constant), the waves go faster. Then the medium is heavy (which we measure in mass per unit length), the waves go slower. I think these things make sense. The equation (which I don’t think you have to memorize) is T v = µ v = wave velocity (m/s) T = Tension (N) µ = mass per unit length (kg/m)

The amplitude of a wave is really about how much energy it’s bringing. It takes more energy to create a large-amplitude wave and a large-amplitude wave can deliver more energy. Amplitude has nothing to do with velocity, frequency or wavelength, it’s separate.

Standing Waves and Musical Instruments (Hewitt 25.8 and 26) When a wave gets to the end of a medium (like the end of a string), it can reflect and go back the other way. If the wavelength “fits” with the length of the medium, this can happen in an organized way that reinforces itself. It’s called a standing wave and is a form of resonance. Here are some standing waves in strings fixed at both ends. These could be strings in any stringed musical instrument like a guitar, violin, or piano.

You should try it! If you get a slinky or a string or something like that (heavier is easier) you should be able to hold it between your hands and shake one hand up and down at different frequencies until you can make some of these shapes. It’s fun!

There is a precise relationship between the length of the string and the wavelength of the wave. And the wavelength of the wave is related to the frequency of the shaking and the speed of the wave in the string.

And we end up with this simple relationship which explains how musical instruments work: nv f = 2ℓ f = frequency (Hz) -- which is the frequency of the musical note n = mode (no units) v = speed of the wave (m/s) ℓ = length of the standing wave (m)

In string instruments, the choice of note is made mostly by changing the length of the string. In the guitar, the player pushes on the string near a fret which stops the vibration of the string there, effectively making the string shorter and the note higher. Strings are tuned by changing the tension like this:

T This uses this equation: v = . Notice that higher tension makes a higher speed which µ makes the frequency higher.

You might also notice that stringed instruments often have strings of different thicknesses. This uses the fact that greater mass per unit length makes the velocity slower and thus the frequency lower. So, the heavier strings are for lower notes.

String instruments rarely use the mode of the standing wave to change the notes. They usually just let all the modes oscillate at once because they sound good together and what we mostly hear is the fundamental.

In wind instruments, very little can be done about the speed of the wave because the medium is the air inside the instrument and the speed is the speed of sound. (The speed of sound does depend on the temperature of the air, that’s why wind-instrument players make sure to warm their instruments up.).

Wind instruments change the length of the standing waves in different ways. The brass instruments (trumpet, trombone, tuba, French horn, etc.) add and subtract lengths of pipe using valves or a slide. The length of the column of air is from the mouthpiece to the bell, through whatever path is opened by the player. The woodwinds have holes that can be covered and uncovered by fingers or keys. The length of the column of air is from the mouthpiece to the first open hole.

Wind instruments also take advantage of the modes of standing waves. By changing what’s happening at the mouthpiece, the player can choose to favor a higher mode than the fundamental and thus get a higher note with the same-length column of air. Brass players make the most use of this, being able to play many notes with each combination of valves or slide position. Woodwinds usually only play the fundamental and first overtone (which is an octave above).

The Doppler Effect (Hewitt 25.9) There seem to be a lot of questions on the SAT about the Doppler Effect. The Doppler Effect describes what happens when the source of the sound is moving and/or when the observer of the sound is moving. When a sound source (like this firetruck siren) is not moving, the sound waves go away from it in the same way in all direction. When the firetruck move, that causes the waves in front of it to get bunched up and the waves behind it to get pulled apart.

If you are Observer A, you are hearing the sound waves bunched up, closer together than they’re supposed to be. This means that the frequency is higher for you than it would be if the firetruck were at rest and higher than what the driver of the truck is hearing (because he is at rest relative to the truck). If you are Observer B, you are hearing the sound waves pulled apart from each other, a lower frequency than they’re supposed to be. So in this case, there are three frequencies being heard. Observer A hears the highest, Observer B hears the lowest, and the driver hears the one in the middle. Make sense?

A similar think happens if the observer is moving. If you move toward the source of sound, you will make the waves bunch up and hear a higher frequency. If you move away, the frequency will be lower.